Truth in advertising

My campus bookstore made a dramatic claim about reduced textbook pricing. Karen Natale, the “Bookstore & Licensing Program Manager” has this to say about the new bookstore and reduced prices:

In case you don’t have time to watch the video, here’s the relevant quote:

As you know, publishers set the prices, the bookstore doesn’t, on [new] textbooks. But bookstores add a margin, and the margin covers overhead, salaries, shipping and handling, all the things we need to do to stay in business. In the past, our margin was 23%, which is what the national average is. Many of the UNC campuses are at 25%. We are now, thanks to our new contract [with Barnes & Noble], at 18%. So that’s a significant savings that students are going to see staying in their pockets.

Wow! A margin of only 18%, when the national average is 23% and some UNC campuses are charging as much as 25%, according to her. Sounds great, right?

Suppose the publisher sells a book to the bookstore for the wholesale price of $105. With an advertised 18% margin, what do you think the shelf price should be? Pause for a moment and calculate this yourself. You probably came up with $123.90. You took $105 and multiplied it by 1.18, right?

Unfortunately, that’s not how it’s done. What you’ve done (and what advertisers are counting on) is to confuse margin with markup. There’s quite a difference.

To be brief, if you start with the sale price, the margin is percentage you *discount* to get the cost, whereas if you start with the cost, the markup is the percentage you *add* to determine the sale price. Thus markup and margin are “dual” to each other in some sense.

To be definite, suppose C is the wholesale cost of the item to the reseller. Let M represent the percent margin or markup desired, with 0\leq M<1. If M represents markup, then the price P of the item is P=C(1+M). If M represents margin, the formula is P=C\frac{1}{1-M} instead. To summarize,

P_\text{markup}=C(1+M)

P_\text{margin}=C\frac{\displaystyle 1}{\displaystyle 1-M}.

So, back to our textbook example. Its wholesale cost to the bookstore was $105.00. Since they have an 18% margin, the price listed on the shelf will be 105\frac{1}{1-.18}=105\frac{1}{.82}=105(1.22)=128.05. Did you notice? An 18% margin became a 22% markup!

In general, if the margin is M then the markup is \frac{M}{1-M}. Here’s a short table showing the relationship in increments of 10%.

0% margin = 0% markup
10% margin = 11.1% markup
20% margin = 25% markup
30% margin = 42.9% markup
40% margin = 66.7% markup
50% margin = 100% markup
60% margin = 150% markup
70% margin = 233% markup
80% margin = 400% markup
90% margin = 900% markup

Assuming 0\leq M<1, we may Taylor expand the margin factor as follows: \frac{1}{1-M}=1+M+M^2+M^3+\cdots. Thus we see that

P_\text{markup}=C(1+M)

P_\text{margin}=C(1+M+M^2+M^3+\cdots)

So the markup formula is really just a first order Taylor approximation to the margin formula. In using the margin formula, they’re benefitting from keeping all those higher order terms, and they can really add up over many, many sales.

What’s worse, the difference between the markup formula and the margin formula becomes more dramatic as the margin goes up. The formulas are identical for M=0, but the margin formula blows up to infinity as M approaches 1 (i.e. 100%). Meanwhile, the honest markup formula remains “honest” no matter how high M goes.

By the way, I don’t place any blame on Karen Natale. She’s been excellent in working with professors and publishers to get low prices and correct errors.

I’m guessing resellers advertise margin instead of markup since, for a fixed profit, the advertised margin will be smaller than the markup, making it sound like you’re getting a better deal than you really are. I suppose this tactic is par for the advertising world. Oh well. Now you know.

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6 Responses to Truth in advertising

  1. Shep says:

    This is by far the most detailed, accurate, and sensible description of the difference between markup and margin. We used to use this concept at Computer Renaissance to advertise deals on used hardware (only we didn’t call it margin).

    Essentially, markup is a variable in calculating margin, but most people confuse the two and assume you’re getting a much better deal.

    Anyway, keep up the good work. I speak very highly of you to everyone I know :).

    • mjfairch says:

      Thanks Shep!

      There’s another way to describe the difference too:

      If you start with the sale price, the margin is percentage you *discount* to get the cost, whereas if you start with the cost, the markup is the percentage you *add* to determine the sale price. Thus they are “dual” to each other in some sense. Maybe I’ll add this to the main body of the post.

      Thanks for browsing my blog!

      Mike

  2. George says:

    A margin of 18% vs. a national average of 23% IS a great deal! Assuming a fixed cost to the retailer, consumers save more from an X% reduction in margin than from an X% reduction in markup, with the difference increasing as the original (non-marked-down) margin increases.

    • mjfairch says:

      Right you are. And indeed it is a good deal … if you’re comparing apples to apples.

      I was implicitly suggesting that spin doctors might compare an 18% margin against a 23% markup, thus fooling an undiscerning consumer into thinking there’s a drastic difference, where in fact there isn’t (since 18% margin = 22% markup).

      Seeing as how I didn’t have any evidence for such a faulty comparison, I didn’t say so explicitly.

      The main point is to draw attention to the difference between markup and margin, which I think is more or less accomplished.

  3. RnS says:

    Very nice writeup!

    One small typo: Let ‘M’ represent the percent margin or markup desired, you have 0 < P < 1 instead of 0 < M < 1.

    (I bumped into your blog after reading an extraordinarily detailed and useful comment of yours on http://skullsinthestars.com/2009/03/18/infinite-series-are-weird/

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